Investigation of the lubrication performance of a marine diesel engine crankshaft using a thermo-electrohydrodynamic model

2.1 Extended Reynolds equation based on average flow model

During the operation of a four strokes diesel engine, for the main bearing lubrication, the crankshaft journal rotates, while the upper and lower bearing shells are constrained and fixed. Therefore, the oil filling rate γ is introduced [11]. Based on the model of average flow, the dynamic load generalized Reynolds equation of the sliding bearing is established, which is solved by the finite difference method.

where x and y are the equation coordinate systems [12]; t indicates the coordinate of time; p indicates the pressure of oil film; h indicates the thickness of oil film [13,14]; υ indicates the journal linear velocity; γ indicates filling rate of oil; and η indicates the lubricating oil dynamic viscosity.

The boundary conditions are set as follows:

1. Axial boundary condition

   (2) P = P a x = ± B 2 ,

   where P a is the environmental pressure; B indicates the bearing width; and p indicates the oil film pressure.

2. Circumferential boundary conditions

   At the oil film starting boundary, immediately γ = 0:

   (3) P = P in ∣ X ∣ ≤ l 2 P = P in B − 2 ∣ X ∣ B − l l 2 < ∣ X ∣ ≤ B 2 ,

   where P in indicates the supply pressure of oil; and l indicates the oil sump width.

   (4) ∂ P ∂ γ = 0 P = P a υ = y r ,

   where r is the radial coordinate.

3. Fuel supply area

   (5) P = P in .

4. Cavity boundary condition

   (6) P = P c γ = 1 P < P c γ < 1 ,

2.2 Asperity contact model

By use of the Greenwood/Tripp micro convex contact theory [15], the contact pressure equation of micro convex body on the bearing surface is

where P b indicates the contact pressure; K indicates the elastic contact factor; σ s indicates the surface roughness; F is the factor of surface roughness; and E ⁎ indicates the comprehensive elastic modulus.

If 4 − h σ s ≥ 0 , we can judge a rough contact area, and in a mixed friction state; if 4 − h σ s < 0 , we can judge a fully lubricated contact area, and in a hydrodynamic lubrication state.

2.3 Oil film thickness equation

Under light load conditions, the deformation of the shells minimally affects the thickness of the oil film. However, in situations of heavy load, the impact of bearing shell and surface deformation on the oil film thickness becomes significant due to the oil film pressure [16]. This study addresses diesel engine performance across diverse conditions and loads, emphasizing the necessity to account for the elastic deformation and surface roughness of the bearing seat, shell back, and anti-friction alloy layer. In solving the generalized Reynolds equation, the roughness of the main bearing journal and shell surface is considered in each iteration step, and the oil film pressure is determined through the finite element method.

2.4 Energy equation

In the general solution, it is assumed that the engine oil viscosity is a constant value under isothermal conditions, but in the process of accurate calculation and actual operation of the diesel engine, thermal effects and other influencing factors must be taken into account, and cavitation effects must also be considered [17]. The main bearing is subjected to alternating loads, and the transient energy equation of its work is shown in Eq. (9), which is solved discretely through the finite element.

where T is the fluid temperature; C p indicates the fluid specific heat capacity at constant pressure; ρ indicates the density of fluid; K indicates the thermal conductivity of the fluid, x , y , and z are three coordinate axis directions, u , v , and w , respectively, denote the velocity of the fluid in the three coordinate axis directions.

Boundary conditions:

1)

where T mix is the mixing temperature in the oil tank.

2)

3) The heat flux at the interface between oil and bearing bush is continuous:

where θ is the oil film equivalent heat transfer coefficient; C b is the specific heat capacity of bearing bush; K b indicates the bearing pad thermal conductivity coefficient; h indicates the bearing pad convection heat transfer coefficient; and R b is the outer diameter of the bearing shell.

4) The journal is regarded as an isothermal body. At the interface between oil and journal

where T s is the journal temperature.

2.5 Heat conduction equation of bearing bush

The diesel engine main bearing in this study is simplified [18]. The solid density is uniform, isotropic, and there is no heat source in the system. The simplified heat conduction equation is

The boundary conditions are as follows:

1)

2)

3) Bearing bush and oil interface

4) Interface between bearing pad and environment

5) On both ends of the bearing

where T a is the ambient temperature.

2.6 Main bearing load balance equation

The force on the journal is decomposed along Y and Z directions, excluding the inertia force of the oil film, the motion of the journal center satisfies Newton’s second law of motion, the equation is as follows:

where R is the journal radius, ω y , ω z are the angular accelerations in y and z directions, m j is the journal mass, F y , F z are the load components in y and z directions, F p y , F p z are the oil film reaction force in y and z directions, respectively.

2.7 Solution scheme of expanded TEHD model

Simultaneous solution of the equation set for extended TEHD model is obtained numerically by using own computer program (Figure 1).

Figure 1

Solution scheme of expanded TEHD model.

3.1 Diesel engine crankshaft-main bearing modeling

This study centers on the crankshaft and main bearing of a marine in-line four-cylinder stroke diesel engine. Initially, a three-dimensional finite element model encompassing the crankshaft, bearing shell, and bearing seat is established. The characteristics of the five main bearings are determined using AVL-Excite software by applying the mathematical equations, yielding essential lubrication parameters [19]. The analysis involves the identification of key lubrication characteristics, such as the maximum oil film pressure and minimum oil film thickness, throughout each load cycle for the five main bearings, providing insights into the main bearing lubrication. Figure 2 illustrates the coupling model of the crankshaft-main bearing multi-body dynamics.

Figure 2

Multi body dynamic coupling model of diesel engine crankshaft main bearing.

In establishing the shafting parameters, essential details such as the diesel engine's operational speed, cylinder count, cylinder bore, stroke, firing sequence, geometric characteristics of the shafting, component quality, gas pressure, and other pertinent factors must be specified. The primary design performance indicators and technical parameters are succinctly presented in Table 1.

Figure 3 illustrates the temporal variation of the cylinder pressure curve at a diesel engine speed of 2,000 rpm, showcasing the fluctuation of cylinder pressure with respect to the crankshaft angle. This curve depicts the cylinder pressure obtained through fitting experimental data provided by the factory.

Figure 3

Cylinder pressure curve of diesel engine.

3.2 Main bearing dynamic characteristics

The crankshaft main journal exhibits periodic movement due to external loads, primarily originating from gas pressure and reciprocating inertia force. As a result, the vibration pattern of the crankshaft relies mainly on the forcing actions exerted by cylinder pressure and reciprocating inertia force.

Figure 4 presents the calculated oil film reactions for each main bearing. The peak oil film reaction values for the five main bearings are 33,720, 45,293, 32,097, 43,041, and 35,985 N, respectively. Among them, the peak values for No. 2 and No. 4 main bearings are the highest, followed by No. 1 and No. 5 bearings. The oil film force peak value of No. 3 main bearing, located in the middle, is the smallest, primarily determined by the relative positions of the firing cylinder and the main bearing. When a cylinder fires, the oil film reaction of adjacent main bearings undergoes a significant change, with impacts propagating from near to far. At a crank angle of 180°, when No. 2 cylinder fires, main bearings No. 2 and No. 3 are most affected, resulting in the maximum oil film reaction for these bearings. The oil film reaction of No. 1 and No. 4 main bearings also fluctuates. The influence on No. 5 main bearing is the least, given its greater distance from the firing cylinder. Additionally, the oil film reaction forces of main bearings 2 and 3 differ due to the inertia forces of adjacent cylinders.

Figure 4

The oil film reaction of main bearing.

3.3 Main bearing lubrication analysis

The lubrication condition of the diesel engine directly affects its operational performance, making continuous research on the lubrication system essential. The optimal lubrication state for the main bearing is hydrodynamic lubrication [20]. The wedge-shaped structure created by the crankshaft and bearing forms a hydrodynamic oil film, which sustains the external load and prevents direct contact between the crankshaft and bearing, effectively reducing friction and wear.

Figure 5 presents a comparison of the maximum oil film pressure in the five main bearings during normal operation. The figure reveals distinct oil film pressure variations for the four working processes of the diesel engine on the crankshaft-bearing. As the engine approaches the third working stroke, the oil film pressure increases, and vice versa. For example, in the case of the 3# main bearing, the maximum crank angle for the oil film pressure occurs at 1,620° CA and 1,980° CA, precisely when the second and third cylinders are performing work, with almost no impact on the oil film pressure in other main bearings.

Figure 5

Comparison diagram of five main bearings maximum oil film pressure.

The comparison of the minimum oil film thickness in the five bearings under normal operation is illustrated in Figure 6. The minimum thickness of the oil film for the first, second, third, fourth, and fifth main bearings are 1.04, 1.23, 1.18, 1.23, and 1 µm, respectively.

Figure 6

Comparison of five main bearings minimum oil film thickness.

The minimum oil film thickness represents the smallest value formed by the diesel engine main bearing during operation. According to the formation mechanism of hydrodynamic lubrication, the minimum thickness of the oil film should be at least five times the surface roughness of the main bearing and main journal. A comparison of the recommended minimum oil film thickness by various internal combustion engine research institutes indicates that the Austrian Liszt Institute requires a minimum main bearing oil film thickness of 1.2–1.4 µm. However, the oil film thicknesses of the 1#, 3#, and 5# main bearings are all lower than this value, suggesting the possibility of contact between the bearing shell and the journal, leading to mixed friction.

4.1 Oil groove width

Theoretical studies by Reynolds on lubrication characteristics indicate that the variation in oil groove width significantly impacts lubrication characteristics. Based on the original oil groove width of 19 mm, dynamic simulations were conducted using three other oil groove widths: 17.6, 18.2, and 19.8 mm.

Figures 7 and 8 illustrate that with an increase in the oil groove width, the minimum thickness of the oil film also increases, while the maximum oil film pressure initially decreases and then increases. The main reason for this behavior lies in the impact of reduced oil groove width on the lubricating oil flow over the bearing shell surface, leading to an elevated load per unit area in the upper and lower directions of the bearing shell. Consequently, during engine operation, the oil flow becomes less smooth, causing a reduction in the minimum oil film thickness. Conversely, an increased oil groove width reduces the load per unit area in the upper and lower directions of the bearing shell, leading to improved lubrication. Sufficient lubricating oil can then lubricate the bearing shell and journal surface, strengthening the oil film lubrication oscillation effect and increasing the minimum oil film thickness. Simultaneously, the peak contact pressure decreases. Nevertheless, considering the overall factors, an appropriate bearing width is crucial for ensuring hydrodynamic lubrication. The ideal groove width should neither be too large nor too small, but rather optimized to achieve the best lubricating oil flow and oil film thickness.

Figure 7

Comparison diagram of 4 # main bearing maximum oil film pressure under different bearing widths.

Figure 8

Comparison diagram of 4 # main bearing minimum oil film thickness under different bearing widths.

4.2 Oil supply pressure

Figures 9 and 10 illustrate the variation in oil supply pressure from 0.3 to 0.6 MPa, while keeping parameters such as oil groove width, journal surface roughness, and bearing clearance unchanged. The maximum pressure of the oil film, minimum oil film thickness, average frictional power consumption, and peak contact pressure between the crankshaft and bearing show little change under these conditions. However, with an increase in oil supply pressure, the lubricating oil flow also increases, which has little effect on the lubrication properties of the crankshaft and bearing. Generally, the normal oil supply pressure for diesel engines can be adequately met. Any pressure drops should be promptly investigated to prevent adverse reactions due to bearing oil shortage.

Figure 9

Comparison diagram of 4 # main bearing maximum oil film pressure under different oil supply pressures.

Figure 10

Comparison diagram of 4 # main bearing minimum oil film thickness under different oil supply pressures.

4.3 Bearing clearance

Figures 11 and 12 show that an increase in the bearing clearance leads to a decrease in the minimum oil film thickness and an increase in the maximum pressure of the oil film. Excessively large clearance results in increased oil leakage from the bearing surface, leading to larger flow and decreased oil film thickness. This, in turn, enlarges the contact area between the journal and bearing, deteriorating the lubrication conditions and making the bearing susceptible to damage, resulting in increased temperature and excessive vibration. Conversely, if the bearing clearance is too small, friction consumption increases, leading to insufficient oil flow and reduced oil film bearing capacity, which also hinders the achievement of effective lubrication. Based on the above analysis, the following conclusions can be drawn:

1. An increase in bearing clearance leads to higher maximum pressure of the oil film and reduced minimum oil film thickness.

2. Excessively small bearing clearance leads to elevated oil temperature and compromised oil film bearing capacity, making it difficult to achieve liquid dynamic lubrication. On the other hand, excessively large bearing clearance can lead to increased lubricating oil leakage and higher loads, hindering effective lubrication.

Figure 11

Comparison diagram of 4 # main bearing maximum oil film pressure under different bearings clearances.

Figure 12

Comparison diagram of 4 # main bearing minimum oil film thickness under different bearing clearances.

4.4 Surface roughness

Figures 13 and 14 demonstrate that an increase in journal surface roughness leads to an increase in the minimum oil film thickness and a decrease in the maximum oil film pressure, which is beneficial for lubrication. However, based on the calculation of the micro contact model, the value of 4 − h σ s in the model is greater than zero, indicating that the main bearing experiences mixed friction. Consequently, a large journal surface roughness promotes the formation of an oil film, but the lubrication state is still not optimal.

Figure 13

Comparison diagram of 4 # main bearing maximum oil film pressure under different roughness.

Figure 14

Comparison diagram of 4 # main bearing minimum oil film thickness under different roughness.